ALBANY — A threat by Gov. Cuomo to keep the Legislature at work through Father’s Day led lawmakers yesterday to agree to extend rent regulations for 2.5 million city tenants at least through Monday at midnight.

“I told them, if the [rent] laws are not in place, I’m going to keep the Legislature in place” through the weekend, said Cuomo, who has the power to call an “extraordinary session” that would force lawmakers to stay at the Capitol.

The regulations, which expired at midnight Wednesday, are the subject of intense negotiations among Cuomo and Assembly Speaker Sheldon Silver (D-Manhattan), who wants to strengthen tenant protections, and Senate Majority Leader Dean Skelos (R-Nassau), who wants to leave existing protections unchanged.

Silver has been insisting that the high-end rent and personal-income thresholds that permit landlords to remove apartments from regulations be lifted, and that the practice of decontrolling some apartments that become vacant be ended.